RF Excitation. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2006 MRI Lecture 4. Thomas Liu, BE280A, UCSD, Fall 2006

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1 Bioengineering 28A Principles of Biomedical Imaging Fall Quarter 26 MRI Lecture 4 RF Excitation From Levitt, Spin Dynamics, 21 1

2 RF Excitation At equilibrium, net magnetizaion is parallel to the main magnetic field. How do we tip the magnetization away from equilibrium? Image & caption: Nishimura, Fig. 3.2 B 1 radiofrequency field tuned to Larmor frequency and applied in transverse (xy) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-axis. - lab frame of reference RF Excitation 2

3 a) Laboratory frame behavior of M b) Rotating frame behavior of M Images & caption: Nishimura, Fig B 1 (t) = 2B 1 (t)cos("t )i (t) cos ("t)i # sin ("t)j ( ) + B 1 (t)( cos ("t)i + sin ("t)j) 3

4 Rotating Frame Bloch Equation dm rot dt = M rot " #B eff B eff = B rot + & ) rot # ; ( + rot = ( + '( %* + Let B rot (t)i + B k B eff = B rot + " rot # % (t)i + B " ( ' * k & # ) If " = " = #B Then B eff (t)i 4

5 Flip angle " = % # 1 (s)ds where # 1 (t) = &B 1 (t) 5

6 slice Slice Selection z Δz f rect(f/w) W=γG z Δz/(2π) sinc(wt) 6

7 Let B rot (t)i + ( B + "G z z)k B eff = B rot + # rot " % (t)i + B + "G z z # ( ' * k & " ) If # = # B eff (t)i + ("G z z)k 7

8 Small Tip Angle Approximation M z M " For small " M xy M z = M cos" # M M xy = M sin" # M " Recall that in the rotating frame, flip angle " = # % B 1 (s)ds Define &(z) as the Larmor Frequency at each location z referenced to & The effective field felt in each spin's rotating frame of reference is: B 1 e (t) (t)exp( j&(z)t) Therefore the flip angle in each spin's frame of reference is t "(t,z) = # % exp( j&(z)s)b 1 (s)ds With respect to the on - resonance frame of reference, there is also a relative phase shift of exp(' j&(z)t), so that t "(t,z) = # exp(' j&(z)t) % exp( j&(z)s)b 1 (s)ds Applying small angle approximation leads to t M r (t,z) ( jm "(t,z) = M # exp(' j&(z)t) % exp( j&(z)s)b 1 (s)ds 8

9 Small Tip Angle Approximation t M r (t,z) = jm exp(" j#(z)t) exp( j#(z)s)# 1 (s)ds For symmetric pulse of length % % / 2 M r (%,z) = jm exp(" j#(z)% /2) exp( j2&f (z)s)# 1 (s + % /2)ds "% / 2 { } f =" f (z)=" ' = jm exp(" j#(z)% /2)F # 1 (t + % /2) 2& G z z B 1 (t) rect t " # /2 ' & ) % # ( # ( ) = jm exp(" j*(z)#) exp( j*(z)s) M r #,z Small Tip Angle Example + * 1 rect s " # /2 ' & ) ds % # ( t = jm exp(" j*(z)# /2)F 1D * 1 rect& ' ' & )) % %# (( ( ) = jm exp(" j*(z)# /2)* 1 # sinc f# = jm exp(" j*(z)# /2)* 1 # sinc &,G z# % 2- z ' ) ( f =" (, / 2- )G z z 9

10 Refocusing M r (3" /2,z) = exp( j#(z)" /2)M r (",z) = jm exp( j#(z)" /2)exp( j#(z)" /2)F{# 1 (t + " /2)} % f = = j M F {# 1 (t + " /2)} % f = 2& G z z 2& G z z Slice Selection RF G z (t) G x (t) Slice select gradient Slice refocusing gradient G y (t) 1

11 Gradient Echo RF G z (t) G x (t) G y (t) ADC Slice select gradient Slice refocusing gradient Spins all in phase at k x= Small Tip Angle Example B 1 (t + " /2) = Asinc t /" & % = Asinc( t /")w(t) '' )) % " (( ( ) cos& 2#t F(B 1 (t + " /2)) = A" rect( f") *W ( f ) = A" rect +G zz" ' & )*W, +G zz' & ) % 2# ( % 2# ( Width of the rect function is "z = 2# G z % 11

12 slice Slice Selection z Δz f rect(fτ) "f = 1 # = G z "z 2% sinc(t/τ) 12

13 Example "z = 5 mm;# = 4 µsec; = %/2 G z = 2% &"z# = 1 =1.175 G /cm (4257Hz /G)(.5cm)(4e ' 6) T ) ( & B 1 sinc s ' T /2, / +. ds ( &B 1 area of sinc * # - ( ) = &B 1 # B 1 = &# = %/2 2%(4257Hz /G)(4e ' 6) =.1468 G 13